AP Statistics
Class Examples Section 10.1 SOLUTIONS!!!
Example 1
Alternate Example: Hearing loss
Are teenagers going deaf? In a study of 3000 randomly selected teenagers in 1988-1994, 15% showed some hearing loss. In a similar study of 1800 teenagers in 2005-2006, 19.5% showed some hearing loss. (Source: Arizona Daily Star, 8-18-2010).
Problem:
(a) Does these data give convincing evidence that the proportion of all teens with hearing loss has increased?
(b) Between the two studies, Apple introduced the iPod. If the results of the test are statistically significant, can we blame iPods for the increased hearing loss in teenagers?
Solution:
(a) State: We will test : = 0 versus : > 0 at the 0.05 significance level where= the proportion of all teenagers with hearing loss in 2005-2006 and = the proportion of all teenagers with hearing loss in 1988-1994.
Plan: We should use a two-sample z test for if the conditions are satisfied.
· Random: The data came from separate random samples.
· Normal: = 351, = 1449, = 450, = 2550 are all at least 10.
· Independent: The samples were taken independently and there were more than 10(1800) = 18,000 teenagers in 2005-2006 and 10(3000) = 30,000 teenagers in 1988-1994.
Do: = 0.167, z = = 4.05, P-value 0
Conclude: Since the P-value is less than 0.05, we reject . We have convincing evidence that the proportion of all teens with hearing loss has increased from 1988-1994 to 2005-2006.
(b) No. Since we didn’t do an experiment where we randomly assigned some teens to listen to iPods and other teens to avoid listening to iPods, we cannot conclude that iPods are the cause. It is possible that teens who listen to iPods also like to listen to music in their cars and perhaps the car stereos are causing the hearing loss.
Alternate Example: Cash for quitters
In an effort to reduce health care costs, General Motors sponsored a study to help employees stop smoking. In the study, half of the subjects were randomly assigned to receive up to $750 for quitting smoking for a year while the other half were simply encouraged to use traditional methods to stop smoking. None of the 878 volunteers knew that there was a financial incentive when they signed up. At the end of one year, 15% of those in the financial rewards group had quit smoking while only 5% in the traditional group had quit smoking. Do the results of this study give convincing evidence that a financial incentive helps people quit smoking? (Source: Arizona Daily Star, 2-11-09).
State: We will test : = 0 versus : > 0 at the 0.05 significance level where= the true quitting rate for employees like these who get a financial incentive to quit smoking and = the true quitting rate for employees like these who don’t get a financial incentive to quit smoking.
Plan: We should use a two-sample z test for if the conditions are satisfied.
· Random: The treatments were randomly assigned.
· Normal: = 66, = 373, = 22, = 417 are all at least 10.
· Independent: The random assignment allows us to view these two groups as independent. We must assume that each employee’s decision to quit is independent of other employee’s decisions.
Do: = 0.100, z = = 4.94, P-value 0
Conclude: Since the P-value is less than 0.05, we reject . We have convincing evidence that financial incentives help employees like these quit smoking.